On the Distribution of a Linear Combination of T-distributed Variables

Abstract:
For odd degrees of freedom the characteristic function of a Student-t random variable is expressible in closed form. The characteristic function of an arbitrary linear combination of independent t-variables is then derived and the distribution function is obtained, itself expressible as a weighted sum of Student-t distribution functions. An easy method of obtaining the weights is demonstrated.
If U1, U1, ..., Un are independent random variables and Xi = d + Ui, i=1,2,...,n are observable random variables, we investigate the choice of a1, a2, ...,an to maximize the power of tests of the form a1X1 + a2X2 + ... + anXn for testing Ho: d = 0 against H1: d > 0. Some general results and examples are given. Of particular interest is the case when Xi is a t-random variable. One application is in a two-stage sampling procedure to solve the Behrens-Fisher problem. The test statistic has the distribution of a weighted sum of t-random variables. It is shown how to choose the weights for maximum power.
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